Let $S$ be a surface in 3D described by the equation $z = 9x^3 - 9y^3 + 5x + 5y$. Fill in the rest of the equation of the plane tangent to $S$ at $(-1, 1)$. $z = $
The equation for a tangent plane of an explicitly defined surface $z = f(x, y)$ at the point $(a, b)$ is: $f(a, b) + f_x(x - a) + f_y(y - b) = z$ [What's the intuition behind the formula?] We can see from the formula that the three values we're missing are $f(-1, 1)$, $f_y$, and the value to subtract from the $x$ -coordinate. The formula should make the $x$ -term zero at $x = -1$, so we want to subtract $-1$ from $x$. $\begin{aligned} &f(-1, 1) = -9 - 9 - 5 + 5 = -18 \\ \\ &f_y = -27y^2 + 5 = -27 + 5 = -22 \end{aligned}$ Here's the completed equation for the tangent plane of $S$ at $(-1, 1)$ : $z = -18 + 32(x - (-1)) + (-22)(y - 1)$